The worm transmissions may be classified into two sorts: the first sort is a cylindrical worm transmission; the second one is a toroidal worm transmission. While the cylindrical worms can further be classified into an involute helicoid worm (ZI for short, the same hereinafter), an Archimedes worm (ZA), a milled helicoid worm (ZK), and a variable tooth worm (VTW) disclosed in China patent No. ZL96244108.2 and U.S. Pat. No. 6,176,148B1 with accordance to the different profile thereof. The worms of ZI-, ZA- and ZK-type can all be formed by lathing with different-profile cutters. However, the relationship of the relative movement between the workpiece and the cutter is the same. Alternatively, the different types of worms such as ZI, ZA and ZK can be machined on the same machine tool with the worm blank turning and the cutter feeding along the longitudinal and radial directions relatively to the worm blank, the one thing must be done only is to change the corresponding cutter. Although the variable tooth worm (VTW) is classified into the cylindrical worms, they cannot be machined on the same facilities with the existing motion relationship even if the cutter were changed seeing that the tooth thickness of the variable tooth worms is changeable along either longitudinal or tooth height direction. The minimum tooth thickness of the worms is at the gorge part of the thread, while it gradually thickens toward both ends, as shown in the FIGS. 1 and 2.
The worms patented as disclosed in U.S. Pat. No. 1,853,643 with the title of “METHOD OF AND APPARATUS FOR GENERATING THE CONVOLUTE TEETH OR THREADS OF WORMS AND THE LIKE” (hereinafter referred to as Simmons patent) are machined by using a gear-shape cutter. The profile of the cutting edges of the gear shaped cutter for machining the thread of worms in Simmons patent is an involute in the end face. Let the radius of the basic cylindroid of the gear shaped cutter be rb, the intersected line of tangential plane Σ with the side surface of the cutter tooth is a straight line as shown in the FIG. 3. Therefore, Simmons method for forming the involute worms can not be used for forming the thread of variable tooth worms with a circular toroid, or an elliptic toroid, or a parabolic toroid.